Presentation on theme: "From clusters of particles to 2D bubble clusters"— Presentation transcript:
1 From clusters of particles to 2D bubble clusters Edwin Flikkema, Simon CoxIMAPS, Aberystwyth University, UK
2 Introduction and overview The minimal perimeter problem for 2D equal area bubble clusters.Systems of interacting particlesGlobal optimisation2D particle clusters to 2D bubble clustersVoronoi construction2D particle systems:-log(r) or 1/rp repulsive potentialHarmonic or polygonal confining potentialsResults
3 2D bubble clusters Minimal perimeter problem: 2D cluster of N bubbles. All bubbles have equal area.Free or confined to the interior of a circle or polygon.Minimize total perimeter (internal + external).Objective: apply techniques used in interacting particle clusters to this minimal perimeter problem.
4 Systems of interacting particles System energy:Usually:Example:Lennard-Jones potential:LJ13: Ar13Used in: Molecular Dynamics, Monte Carlo, Energy landscapes
5 Energy landscapes Energy vs coordinate Stationary points of U: zero net force on each particleMinima of U correspond to (meta-)stable states.Global minimum is the most stable state.Local optimisation (finding a nearby minimum) relatively easy:Steepest descent, L-BFGS, Powell, etc.Global optimisation: hard.Energy vs coordinateLocal optimisation
6 Global optimisation methods Inspired by simulated annealing:Basin hoppingMinima hoppingEvolutionary algorithms:Genetic algorithmOther:Covariance matrix adaptionSimply starting from many random geometries
7 2D particle systems Energy: Repulsive inter-particle potential: Confining potential:orharmonicpolygonal
8 2D particle clustersPictures of particle clusters: e.g. N=41, bottom 3 in energy
10 2D particle clusters Polygonal confining potential: e.g. triangular unit vectorscontour linesdiscontinuous gradient: smoothing needed?
11 Technical details List of unique 2D geometries produced Problem: permutational isomers.Distinguishing by energy U not sufficient:Spectrum of inter-particle distances compared.Gradient-based local optimisers have difficulty with polygonal potential due to discontinuous gradientSmoothing needed?Use gradient-less optimisers (e.g. Powell)?
14 ConclusionsOptimal geometries of clusters of interacting particles can be used as candidates for the minimal perimeter problem.Various potentials have been tried. 1/r seems to work slightly better than –log(r).Using multiple potentials is recommended.Polygonal potentials have been introduced to represent confinement to a polygon
16 Energy landscapes Energy vs coordinate Stationary points of U: zero net force on each particleMinima of U correspond to (meta-)stable states.Global minimum is the most stable state.Saddle points (first order): transition statesNetwork of minima connected by transition statesLocal optimisation (finding a nearby minimum) relatively easy:L-BFGS, Powell, etc.Global optimisation: hard.Local optimisationEnergy vs coordinate
17 2D clusters: perimeteris fit to data for free clusters